Applying bias to the diaphragm does result in a charge induced negative tension increment.
I posted some plots of this behavior several years ago when I didn't understand what was going on.
arend-jan was kind enough to point me to the Peter Baxandall chapter on ESLs which discusses it in detail.Diaphragm Resonance change with HV bias

Taking a look at the negative stiffness term of the force equation....
F= Eps0*A*2*Vbias²*x/d³

You can see that it increases as square of Vbias, but decreases as cube of D/S spacing. So, for the same panel area and diaphragm tension, if you double the D/S spacing and double the Vbias value to obtain the same efficiency, the negative stiffness term will have less affect on Fs.

In your case you have small areas due to segmentating the diaphragm into smaller sections, and your D/S is fairly large(3-3.5mm I think you had said). So, you may see only very small changes in Fs with bias change until you start getting above 8kV or so.